Recent developments have made it possible to consider high-temperature superconductor (HTS) for the design of tokamak toroidal field (TF) magnet systems, potentially influencing the overall design and maintenance scheme of magnetic fusion energy devices. Initial assessments of the engineering challenges and cryogenic-dependent cost and parameters of a demountable, HTS TF magnet system have been carried out using the Vulcan tokamak conceptual design (R=1.2m, a=0.3m, B0=7T) as a baseline. Jointed at the midplane to allow vertical removal of the primary vacuum vessel and routine maintenance of core components, structural D-shaped steel support cases provide cryogenic cooling for internally routed YBCO superconducting cables. The cables are constructed by layering ∼50μm thick commercially available YBCO tape, and the interlocking steel support cases self align during assembly to form internal resistive joints between YBCO cables. It is found that designing the TF magnet system for operation between 10K and 20K minimizes the total capital and operating cost. Since YBCO is radiation-sensitive, Monte Carlo simulation is used to study advanced shielding materials compatible with the small size of Vulcan. An adequate shield is determined to be 10cm of zirconium borohydride, which reduces the nuclear heating of the TF coils by a factor of 11.5 and increases the YBCO tape lifetime from two calendar years in the unshielded case to 42 calendar years in the shielded case. Although this initial study presents a plausible conceptual design, future engineering work will be required to develop realistic design solutions for the TF joints, support structure, and cryogenic system.